# $\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains

I am studying the heat equation on a general bounded domain $$\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$$ with continuously differentiable Dirichlet data $$\phi$$ on the boundary,

$$\left\{ \begin{array}{ccc} \partial_t u - \Delta u = 0 & \text{in } \Omega\\ u = \phi & \text{ on } \partial_n \Omega \end{array} \right.$$

Here $$\partial_n \Omega$$ denotes the normal boundary of $$\Omega$$, i.e. the points of the boundary on which one can solve the Dirichlet problem (with the Perron method for instance, see Watson's book [2]) for any $$\phi \in \mathcal{C}(\partial_n \Omega)$$. It essentially consists of points of the boundary $$\partial \Omega$$ which are not part of the "top" or "cap", more precisely any lower half ball centered at some point $$X_0\in \partial_n \Omega$$ meets the complementary of $$\Omega$$.

I want to derive a fine gradient bound on the solution $$u$$ on this normal boundary $$\partial_n \Omega$$. I don't seem to find better estimates than the one given in the book of Gary Lieberman [1]. He finds that for domains $$\Omega$$ with particular structure conditions, one can obtain,

$$\sup_{|X-X_0| \neq 0} \frac{u(X) - u(X_0)}{|X-X_0|} \leq C$$

His arguments mainly rely on finding sub and supersolutions in the form $$w = \phi + f(d)$$ where $$d$$ denotes the distance to some simple domain $$D$$ containing $$\Omega$$ and $$f$$ an increasing and concave real-valued function. These sub/supersolutions agree with $$u$$ at $$X_0$$ so one can bound $$u$$ above and below through the maximum principle. But nothing more precise can be obtained with this method, and I have found nothing in the literature giving more precise gradient bounds.

I have the intuition that for $$\mathcal{C}^2$$ domains with bounded mean curvature, we can have $$u \in \mathcal{C}^1(\overline{\Omega})$$, meaning that the total gradient $$Du$$ has a well defined limit on the boundary of the domain.

I have tried studying the problem satisfied by $$\partial_i u$$ in $$\Omega$$ without much success : the boundary data is not well defined, not continuous a priori.

I would really appreciate any input on the issue. Ideas or papers/books related to this question. Thank you for your reading.

References

[1] Gary M. Lieberman, Second order parabolic differential equations (English), Singapore: World Scientific Publishers, pp. xi+439 (1996), ISBN: 981-02-2883-X, MR1465184, Zbl 0884.35001.

[2] Neil A. Watson, Introduction to heat potential theory (English), Mathematical Surveys and Monographs 182. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4998-9/hbk), xiii, 266 p. (2012), MR2907452, Zbl 1251.31001.

• I edited the question, $\phi$ is assumed to be continuously differentiable. Jan 28, 2022 at 8:48
• So, you are considering a normal parabolic problem but instead of considering a problem on a cylinder, $\Omega=(0,T)\times \Omega$ you are considering a problem on a general smooth domain in time and space and that is where the problem resides. It isn't really the heat equation in that case, though.. Jan 28, 2022 at 11:58
• The differential operator still is the heat operator. Could you come up with a better name ? Actually the solvability of the Dirichlet problem on time evolving domains is well known (see the references), however I want more precise estimates near the boundary on the domain. Jan 28, 2022 at 15:11
• The estimates for the standard (cylinder) problem are in Evans' PDE textbook. You can use the semigroup approach... Jan 28, 2022 at 20:02
• This approach does not easily generalise for non-cylindrical domains, it is almost exclusively designed for analysis on cylinders. Thank you for your input though. Feb 7, 2022 at 15:56